Stieltjes polynomials and Critical measures

In 1885 Stieltjes introduced certain class of polynomials defined as solutions of a second order linear differential equation with polynomial coefficients. Zeros of those polynomials were interpreted as the minimum of a discrete energy functional. We consider more general situation where zeros of Stieltjes polynomials present a critical point (saddle point) of energy. We introduce and investigate “continualization” of the problem which leads to a general concept of a critical measure. Case related to Stieltjes polynomials turns to be closely related to many other classical concepts and problems (global structure of trajectories of quadratic differentials, moduli problems, extremal domains, S-curves and so on).